Higher Sobolev regularity for the fractional $p-$Laplace equation in the superquadratic case

We prove that for $p\ge 2$ solutions of equations modeled by the fractional $p-$Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in $W^{1,p}_{loc}$ and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation $s$ reaches $1$.